Method and apparatus for secure digital chaotic communication

ABSTRACT

A method and apparatus that uses control of a chaotic system to produce secure digital chaotic communication. Controls are intermittently applied by a transmitter-encoder to a chaotic system to generate the 0 and 1 bits of a digital message. A new control/no control bit stream is thereby created in which a 0 indicates that no control was applied and a 1 indicates that a control was applied. The control/no control bit stream and a prepended synchronization bit stream are transmitted, using conventional transmission technologies, from the transmitter-encoder to an identical receiver-decoder. A chaotic system in the receiver-decoder is driven into synchrony and is subject to intermittent controls based on the control/no control bit stream, thereby causing it to generate the digital message.

STATEMENT OF RELATED CASES

This application claims the benefits of U.S. Provisional Application No.60/107,937 filed Nov. 12, 1998.

GOVERNMENT RIGHTS

The present invention was made with United States Government supportunder Contract No. 93-F152600-00 awarded by the Office of Research andDevelopment. The United States Government has certain rights therein.

FIELD OF THE INVENTION

The present invention relates generally to a method and apparatus forsecure digital chaotic communication. More specifically, it relates to asystem for encoding and decoding information by controlling a chaoticsystem.

BACKGROUND OF THE INVENTION

Secure communication is employed for maintaining both the authenticityand the confidentiality of information. There are many different systemsfor secure communication currently available. However, on the one hand,increases in computing power have raised questions about the security ofmany of these systems, and, on the other hand, the more secure systemsare so complicated that the speed of processing is becoming a limitingfactor in transmitting information. Secure communication based onchaotic systems is a rapidly developing field of research. In general, achaotic system is a dynamical system which has no periodicity and thefinal state of which depends so sensitively on the system's preciseinitial state that its time-dependent path is, in effect, long-termunpredictable even though it is deterministic.

One approach to the use of chaotic systems to encode informationrequires the transmission of a key to decode the information. One suchmethod of secure communication uses a chaotic equation to produce randomnumbers. Banco U.S. Pat. No. 5,048,086. In summary, this approachconverts a sequence of numbers produced by a chaotic equation intodigital form, adds the converted numbers to the digital message to beencoded and transmits the combined digital stream to a receiver. Thereceiver extracts the digital message from the transmitted digitalstream by generating the same sequence of digital numbers using the samechaotic equation as the key. See Weiss U.S. Pat. No. 5,479,512. Thedisadvantage of this approach is the decreased security resulting fromthe transmission of the key.

Other approaches to the use of chaotic systems for secure communicationdo not require the transmission of a key. One such approach involves thesynchronization of chaotic systems. Carroll U.S. Pat. No. 5,473,694 andCuomo U.S. Pat. No. 5,291,555. A parameter of a chaotic signal ismodulated with an information bearing signal or an information bearingsignal is added to a chaotic signal. The resulting chaotic signal istransmitted, using conventional transmission technologies, from atransmitter-encoder to an identical receiver-decoder. Thereceiver-decoder is driven into synchrony by the original chaotic signalwith no key exchange necessary. Comparison of the information bearingchaotic signal is made with the synchronization signal to extract theoriginal information. However, in these systems, if the transmission isintercepted, it is possible to use phase space reconstruction toreconstruct the underlying dynamic of the transmitter-encoder. In somecases, it has been shown that the ability to reconstructtransmitter-encoder dynamics makes it possible to extract theinformation bearing signal using non-linear dynamic (“NLD”) forecastingor other technologies.

Another approach to secure communication is to “control” a chaoticsystem by applying very small perturbations to the system. [S. Hayes, C.Grebogi, E. Ott, and A. Mark, Experimental Control of Chaos forCommunication, Phys. Rev. Lett. 73, 1781 (1994)] To summarize thecontrol approach, a transmitter-encoder encodes a signal by controllingthe sequence of output peaks of a chaotic oscillator through theapplication of small amplitude perturbations to the oscillator. Areceiver-decoder extracts the signal by observing the sequence of peaksof the transmitted signal. However, analysis of this communicationsystem through reconstruction of the phase space dynamics or by NLDforecasting detects the imposition of controls. Thus, a chaotic systemthat transmits a signal that can be used to reconstruct the underlyingchaotic system has limited security.

It is an object the present invention to use “control” of a chaoticsystem for a secure communication system. It is a further object to doso with a transmitted signal that cannot be used for reconstruction ofthe underlying chaotic system, thereby thwarting attempts to reveal thetransmitted message by reconstruction techniques or NLD forecasting. Itis also a further object of the present invention to provide a means bywhich the transmitter-encoder and receiver-decoder of such a securecommunication system are synchronized.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a new method andapparatus for secure digital chaotic communication. Another object ofthe present invention is to provide a faster, more secure method andapparatus for digital communication by controlling a chaotic system.

The present invention may be implemented either in hardware or software.Controls are intermittently applied by a transmitter-encoder to achaotic system to generate a sequence of bits 0 and 1 corresponding tothe plaintext of a digital message. A control/no control bit stream isthereby created in which a 0 indicates that no control was applied and a1 indicates that a control was applied. The control/no control bitstream and a prepended synchronization bit stream are transmitted, usingconventional transmission technologies, from the transmitter-encoder toan identical receiver-decoder. A chaotic system in the receiver-decoderis driven into synchrony and is then subject to intermittent controlsbased on the control/no control bit stream, thereby causing it togenerate the digital message.

The foregoing and other objects, features and advantages of the currentinvention will be apparent from the following more detailed descriptionof preferred embodiments of the invention as illustrated in theaccompanying drawings.

IN THE DRAWINGS

FIG. 1 is a block diagram of a digital chaotic communication systemaccording to an embodiment of the present invention.

FIG. 2 is a flow chart showing the encoding and decoding procedures ofthe digital chaotic communication system shown in FIG. 1.

FIG. 3 is a plot of the double scroll oscillator resulting from thegiven differential equations and parameters.

FIG. 4 is a plot of the symbolic dynamics function, r(x).

FIG. 5 is a plot of the Poincare Map for the given double scrolloscillator.

FIG. 6 is a histogram of the mapping between 4-bit messages and 4-bittransmissions.

DETAILED DESCRIPTION. OF THE INVENTION

FIG. 1 shows a secure digital chaotic communication system 10 accordingto an embodiment of the present invention. A plaintext message is inputto a transmitter-encoder 12 that encodes and transmits a digital chaoticcommunication over transmission media 14. The digital chaoticcommunication is received and decoded by a receiver-decoder 16, and theplaintext message is output.

FIG. 2 is a flow chart of the method and apparatus of the secure digitalchaotic communication of the present invention. The encoding ofinformation involves four steps, 20, 22, 24 and 26. The first step 20for the encoding of information is to input a message bit stream to thetransmitter-encoder 12. Information may be received for inputting from avariety of sources, including from transmissions; from a variety ofinput/output devices associates with the transmitter-encoder 12, such asa keyboard or mouse; from a card read/write device; or from othersources of information known in the art.

The second step 22 for the encoding of information is for thetransmitter-encoder 12 to apply controls to a first chaotic system. In apreferred embodiment, the chaotic system is a double-scroll oscillator[S. Hayes, C. Grebogi, and E. Ott, Communicating with Chaos, Phys, Rev.Lett. 70, 3031 (1993)], described by the differential equations

C1{dot over (v)} _(C1) =G(v _(C2) −v _(C1))−g(v _(C1))

C2{dot over (v)} _(C2) =G(v _(C1) −v _(C2))+i _(L)

L{dot over (i)} _(L) =−v _(C2),

where

${g(v)} = \left\{ \begin{matrix}{{m_{1}v},} & {{{{if} - B_{p}} \leq v \leq B_{p}};} \\{{{m_{0}\left( {v + B_{p}} \right)} - {m_{1}B_{p}}},} & {{{{if}\quad v} \leq {- B_{p}}};} \\{{{m_{0}\left( {v - B_{p}} \right)} + {m_{1}B_{p}}},} & {{{if}\quad v} \geq B_{p}}\end{matrix} \right.$

The attractor that results from a numerical simulation using theparameters C1, [C₁]={fraction (1/9)}, C2[C₂]=1, L={fraction (1/7)},G=0.7, m₀=−0.5, m₁=−0.8, and B_(p)=1 has two lobes, each of whichsurrounds an unstable fixed point, as shown in FIG. 3.

Because of the chaotic nature of this oscillator's dynamics, it ispossible to take advantage of sensitive dependence on initial conditionsby carefully choosing small perturbations to direct trajectories aroundeach of the loops of the attractor. In this way, a desired bit streamcorresponding to the message bit stream can be generated by steering thetrajectories around the appropriate lobes of the attractor, suitablylabeled 0 and 1.

There are a number of means to control the chaotic oscillator to specifythe bits 0 and 1 more precisely. In a preferred embodiment, a Poincaresurface of section is defined on each lobe by intersecting the attractorwith the half planes i_(L)=±GF, |v_(C1)|≦F, where F=B_(p)(m₀−m₁)/(G+m₀).

When a trajectory intersects one of these sections, the correspondingbit can be recorded. Then, a function r(x) is defined, which takes anypoint on either section and returns the future symbolic sequence fortrajectories passing through that point. If 1₁, 1₂, 1₃, . . . representthe lobes that are visited on the attractor (so 1_(i) is either a 0 or a1), and the future evolution of a given point x₀ is such that x₀→1₁, 1₂,1₃, . . . , 1_(N) for some number N of loops around the attractor, thenthe function r(x) is chosen to map x₀ to an associated binary fraction,so r(x₀)=0.1₁1₂1₃. . . 1_(N), where this represents a binary decimal(base 2). Then, when r(x) is calculated for every point on thecross-section, the future evolution of any point on the cross-section isknown for N iterations. The resulting function is shown in FIG. 4, wherer(x) has been calculated for 12 loops around the attractor.

Control of the trajectory begins when it passes through one of thesections, say at x₀. The value of r(x₀) yields the future symbolicsequence followed by the current trajectory for N loops. If generationof the desired message bit stream requires a different symbol in the Nthposition of the sequence, r(x) can be searched for the nearest point onthe section that will produce the desired symbolic sequence. Thetrajectory can be perturbed to this new point, and it continues to itsnext encounter with a surface. This procedure can be repeated until theentire message bit stream has been produced. It should be noted thatthis embodiment exhibits a “limited grammar,” which means that not allsequences of 0's and 1's can be directly encoded, because the chaoticoscillator always loops more than once around each lobe. Consequently, asequence of bits such as 00100 is not in the grammar since it requires asingle loop around the 1-lobe. A simple remedy is to repeat every bit inthe code or append a 1- or 0-bit to each contiguous grouping of 1- or0-bits, respectively. Other embodiments may have a different grammar,and examples exist where there are no restrictions on the sequence of0's and 1's. For this system, the actual transmitted signal is thecoordinate i_(L), so the message bit stream is read from the peaks andvalleys in i_(L) (there are small loops/minor peaks that occur as thetrajectory is switching lobes of the attractor, but these are ignored).An important point to note is that the perturbation is done at constanti_(L) so there is no discontinuity in the transmitted trajectory.

The third step 24 for the encoding of information is for thetransmitter-encoder 12 to create a control/no control bit stream basedon its application of intermittent controls. The control/no control bitstream allows a second chaotic system in the receiver-decoder 16 toremain synchronized with the first chaotic system in thetransmitter-encoder, without transmitting i_(L). To do this, use is madeof the r(x) function, which can be determined independently and taken asknown information for the transmitter-encoder and receiver-decoder. Ther(x) function is secret information unrelated to any of the informationto be passed from the transmitter-encoder to the receiver-decoder. Solong as both the transmitter-encoder and the receiver-decoder haveknowledge of r(x), if the two systems are initially synchronized, thenall that must be maintained for the synchrony to be preserved is for thetransmitter-encoder to tell the receiver-decoder when it has applied acontrol, under the assumption that the control moves the trajectory ofthe chaotic system to the nearest location that gives the desired symbolsequence.

The calculation of r(x) in the preferred embodiment was done discretelyby dividing up each of the cross-sections into 2001 partitions (“bins”)and calculating the future evolution of the central point in thepartition for up to 12 loops around the lobes. As an example, controlswere applied so that effects of a perturbation to a trajectory would beevident after only 5 loops around the attractor. In addition torecording r(x), a matrix M was constructed that contains the coordinatesfor the central points in the bins, as well as instructions concerningthe controls at these points. These instructions simply tell how far toperturb the system when it is necessary to apply a control. For example,at an intersection of the trajectory with a cross-section, if r(x₀)indicates that the trajectory will trace out the sequence 10001, andsequence 10000 is desired, then a search is made for the nearest bin tox₀ that will give this sequence, and this information is placed in M.(If the nearest bin is not unique, then there must be an agreement aboutwhich bin to take, for example, the bin farthest from the center of theloop.) Because the new starting point after a perturbation has a futureevolution sequence that differs from the sequence followed by x₀ by atmost the last bit, only two options need be considered at eachintersection, control or no control. Consequently, when the firstchaotic system of the transmitter-encoder is being perturbed to traceout a given message, the set of controls that are applied can betranslated into another digital sequence, and the map between a stringof message bits and the associated digital sequence of controls changesas a function of both the history of the transmitter-encoder's firstchaotic system and the message bit stream.

Because both the transmitter-encoder and the receiver-decoder havecopies of r(x) and the matrix M, and a protocol has been established sothat the receiver-decoder knows where to start applying the controls,all that need be transmitted from the transmitter-encoder to thereceiver-decoder for the communication of a message bit stream is thesequence of controls in digital form, telling the receiver-decoder whento perturb the trajectory. The matrix M holds the information aboutwhich bin should hold the new starting point for the perturbedtrajectory, so once the receiver-decoder is told to perturb the orbit,it immediately knows where and how to achieve the desired perturbation.In an analog hardware implementation of the preferred embodiment, theperturbations are applied using voltage changes or current surges; in amapping-based hardware implementation, the perturbations are containedin a look-up table and would result in a variable replacement in themapping function. In a software implementation of the preferredembodiment, the control matrix M would be stored along with the softwarecomputing the chaotic dynamics so that when a perturbation is required,the information would be read from M.

As the transmitter-encoder applies intermittent controls to the chaoticsystem to trace out the desired trajectory, at each intersection it isnoted whether or not it is necessary to perturb the system. A 1indicates that a control was applied to perturb the system, and a 0means that the trajectory was allowed to pass through the sectionunperturbed. This control/no control bit stream now forms the signal tobe transmitted. The signal to be transmitted is thus a digital stream,which should have the added benefit of producing a more robustcommunication technique through the use of current hardware anderror-correction technology. The receiver-decoder has a second chaoticsystem that is identical to the first chaotic system, along with acopies of M and r(x), so all the receiver-decoder needs is a startingpoint and the control/no control bit stream. As the trajectory of thereceiver-decoder's second chaotic system passes through some prescribedbin, or, as described in more detail below, after it has been drivenonto a periodic orbit, controls based on the control/no control bitstream are applied. The second chaotic system in the receiver-decoder isthen controlled to follow the same dynamics as the first chaotic systemin the transmitter-encoder and the message bit stream can be read simplyby observing the sequence of lobes of the attractor visited by thesecond chaotic system in the receiver-decoder.

In the sample below, a given message bit stream is shown. Beneath themessage bit stream is a control/no control bit stream produced by thepreferred embodiment of the present invention, which was transmitted tothe receiver-decoder, followed by the recovered message bit stream atthe receiver.

Message: 0011001111000110011110001111100 . . .

Transmitted: 1010101000100001010011001001010 . . .

Recovered: 0011001111000110011110001111100 . . .

The message recovery is exact, and the transmitted control/no controlbit stream bears no obvious relationship to the message bit stream.

A further improvement involves the use of microcontrols. Each time atrajectory of the transmitter-encoder's first chaotic system orreceiver-decoder's second chaotic system passes through a cross-section,the simulation is backed-up one time step, and the roles of time andspace are reversed in the Runge-Kutta solver so that the trajectory canbe integrated exactly onto the cross-section without any interpolation.Then, at each intersection where no control is applied, the trajectoryis reset so that it starts at the central point of whatever bin it isin. This resetting process can be considered the imposition ofmicrocontrols. It removes any accumulation of round-off error andminimizes the effects of sensitive dependence on initial conditions,effectively making the communication technique more robust. It also hasthe effect of restricting the dynamics of the transmitter-encoder to afinite subset of the full chaotic attractor although the dynamics stillvisit the full phase space. These restrictions can be relaxed bycalculating r(x) and M to greater precision at the outset.

Another embodiment of the present invention utilizes an approximateone-dimensional Poincare map. The Poincare section has two branches, oneon each lobe of the attractor. The partitioning of the surface and theuse of microcontrols allow one to calculate easily a map that exhibitsall of the symbolic dynamics of the full microcontrolled system. Theevaluation of this map is much simpler and faster than integratingbetween intersections with the surface of section. To find the map, onecan take the center point in each bin as an initial condition (sincethese are the points to which the micro controls “reset” trajectories),integrate forward in time until the next intersection with either branchof the surface of section, and note the branch and bin in which thetrajectory landed. For a given set of integration parameters (time step,method, etc.) and for a given partition of the surface of section, thetrajectory from the center of any bin to its next intersection with thesurface will not vary. Therefore, the map mimics exactly the behavior ofthe microcontrolled system for the given integration method.

To implement this map, two more columns are placed in the instructionmatrix M: one containing the row number in M that corresponds to thenext intersection for all 2001 bins, and the other containing the nextlobe under the map. Simulated data transmission and reception using thisnew matrix is essentially the same as transmission and reception usingintegration. For a given bin on the section and for a given message bit,the transmitter-encoder still uses the function r(x) to compare thesymbolic dynamics N bits in the future. If the N-th bit in the futuredynamics for that bin differs from the current message bit, r(x) is usedto find the nearest bin that will produce the desired sequence, and a 1is sent. Otherwise, a 0 is sent. Then the map is used to find thelocation of the next intersection with the surface, and the process isrepeated with the next message bit. The use of this map eliminatestime-consuming numerical integration, allowing for faster and moreextensive processing.

The above map differs from a conventional Poincare map in a couple ofaspects. First, while the Poincare section is two-dimensional, it isbeing approximated with a pair of lines extending from the unstablefixed points fitted with a least-squares method. Whenever a trajectoryintersects the section, by only considering the distance from thecorresponding fixed point, the point of intersection is essentiallyrotated about the fixed point onto the line before proceeding. Thereforethe three-dimensional dynamical system is reduced to a one-dimensionalmap. Secondly, the point is reset to the center of its current bin tosimulate the microcontrols. Theoretically, letting the maximum length ofthe intervals in the partition go to zero would make this secondapproximation unnecessary. A plot of the map derived above is shown inFIG. 5. The primarily unimodal shape is not surprising since unimodalmaps exhibit chaotic properties.

The previously discussed reduction of the digital chaotic communicationsystem to one that uses a Poincare map allows a generalization of thesystem to any chaotic one-dimensional map. It is simply a matter ofdefining “lobes”—what section of the domain implies a switching of bits,recording the symbolic dynamics in r(x) and finding appropriate controlsas before. For example, one could take the logistics mapx_(n)=ax_(n−1)(1−x_(n−1)) and somewhat arbitrarily say that for anyx_(k)≧x_(lobe), where 0<x_(lobe)<1, the current bit b_(k) will be1−b_(k−1): -otherwise, b_(k)=b_(k−1). This gives the symbolic dynamicsnecessary to build a system, which can be improved in at least two ways.First, maps can be chosen that would have little to no grammarrestriction, which would eliminate the need to adjust the message bitstream to comply with the system's dynamics. Second, it may be possibleto fine-tune the maps to optimize the system statistically (eliminatingthe grammar restrictions in many ways helps to improve the statistics).

To eliminate the restriction that bits must at least come in pairs, itis necessary that the map allow trajectories to remain in the“switching” region for two or more iterations in a row. For example, onecan use the second iterate of the logistics map,x_(n)=a²x_(n−1)(1−x_(n−1))(1−ax_(n−1)(1−x_(n−1))), with a=3.99. Topreserve the symmetry, it is logical to choose x_(lobe)=0.5. All shortN-bit words are possible in the natural evolution of this map, at leastfor N<4, so there are no grammar restrictions with this system.Therefore, preprocessing any message bit stream is unnecessary.

Another improvement on the present invention, as described above,involves cascading transmitter-encoder stages and receiver-decoderstages. A message bit stream is first passed through atransmitter-encoder to produce a control/no control bit stream. It canbe fed into a second level transmitter-encoder to further transform thebit stream, and so on through a cascade of transmitter-encoders. Thefinal output bit stream is then transmitted to a receiver-decodercascade. At the receiver end, the received bit stream is sent into thelowest level receiver-decoder, and the output is passed through the nextlevel receiver-decoder to move one level higher in the receiver cascade.Once the received bit stream is passed through the same number of levelsas are in the transmitter-encoder cascade, the original message bitstream is recovered. The double-scroll system has a limited grammar;consequently, between any two levels in the cascade, the bit streams arepre-processed to handle the single 1's or 0's before passing to the nextlevel in the cascade. With other systems, preprocessing may beunnecessary.

The process of cascading chaotic transmitter-encoders andreceiver-decoders is illustrated by the results in the following table,showing two levels in the cascade. There are two intermediate steps ofpreprocessing at the transmitter-encoder end where extra 1's and 0's areappended to contiguous sections of 1's and 0's in the bit stream thatwill be fed into each level of the cascade. At the receiver-decoder end,the reverse processing must be applied, where a bit must be strippedfrom each contiguous group of 1's and 0's in the bit stream:

Original Message: 00111011001100001111101011100101110011110110010000 . ..

Pre-processed Message: 0001111001110001110000011111100110011110001100111. . .

Intermediate Output: 00000011010010010000000100001101010100010000101001. . .

Pre-processed Intermediate:00000001110011000110001100000000110000011100110011 . . .

Transmitted Bit Stream:00010011001010100101000010010000101000010010101010 . . .

Received Level1: 00000001110011000110001100000000110000011100110011 . ..

Intermediate Output: 00000011010010010000000100001101010100010000101001. . .

Received Level2: 00011110011100011100000111111001100111100011001111 . ..

Recovered Message: 00111011001100001111101011100101110011110110010000 .. .

The message recovery is exact and the transmission bit stream bears noobvious correlation with the message bit stream.

The next step 26 in the preferred embodiment of the present invention isthe addition by the transmitter-encoder of a synchronization bit streamto the control/no control bit stream, thereby creating a transmissionbit stream. The synchronization bit stream allows the first chaoticsystem in the transmitter-encoder and the second chaotic system in thereceiver-decoder to synchronize initially. It is possible to send asequence of controls to the second chaotic system in thereceiver-decoder that will drive it onto a periodic orbit. Once on theperiodic orbit, the message bits can be incorporated into the dynamicsof the transmitter-encoder, with the resulting transmitted bits causingthe second chaotic system of the receiver-decoder to leave the periodicorbit, which serves to alert the receiver-decoder to the beginning ofthe message.

At a fundamental level, when microcontrols are used in the digitalcommunication system, there are only a finite number of orbits on theattractor, so periodicity of a chaotic system would eventually beguaranteed under a repeating sequence of controls. More importantly, thesecond chaotic system in the receiver-decoder can be driven onto aperiodic orbit by sending it a repeating code. Different repeating codeslead to different periodic orbits. The periodic orbit reached isdependent only on the code segment that is repeated, and not on theinitial state of the second chaotic system in the receiver-decoder(although the time to get on the periodic orbit can vary depending onthe initial state). Consequently, it is possible to send aninitialization control sequence to the receiver-decoder, that drives thesecond chaotic system in the receiver-decoder and the first chaoticsystem in the transmitter-encoder onto the same periodic orbit.

There are numerous control sequences that, when repeated, lead to aunique periodic orbit for all initial states, so that there is aone-to-one association between a sequence and the orbit. However, forsome control sequences the orbits themselves change as the initial stateof chaotic system changes. Consequently, repeated control sequences canbe divided into two classes, initializing and non-initializing. Thelength of each periodic orbit is an integer multiple of the length ofthe repeated control sequence. This is natural, since periodicity isattained only when both the current position on the cross-section aswell as the current position in the control sequence is the same as atsome previous time. To guarantee that the first chaotic system in thetransmitter-encoder and second chaotic system in the receiver-decoderare synchronized, it is sufficient that the period of the orbit isexactly the length of the smallest repeated segment of the initializingcontrol sequence. Otherwise, since the control sequence is the only linkbetween the transmitter-encoder and the receiver-decoder it is possiblethat the first chaotic system in the transmitter-encoder and the secondchaotic system in the receiver-decoder could be on the same periodicorbit, yet out of phase. Consequently, the first chaotic system and thesecond chaotic system would not be truly synchronized.

Step 30 in the Flow chart in FIG. 2. is the transmission of thetransmission bit stream using conventional transmission technologies.Steps 40, 42, and 44, which comprise the decoding process, are asdescribed above, basically the inverse of the encoding process of steps22, 24 and 26. Step 50 is the output of the message bit stream.

The method and apparatus of the present invention can be implementedentirely in software. The chaotic systems in the transmitter-encoder andreceiver-decoder in such an implementation are defined by a set ofdifferential equations governing the chaotic dynamics, e.g., the doublescroll equations described above. The transmitter-encoder andreceiver-decoder utilize the same algorithm to simulate the evolution ofthe differential equations, e.g., the fourth order Runge-Kuttaalgorithm. The:transmitter-encoder and receiver-decoder also holdduplicate copies of a file that contains all of the control informationthat is necessary to maintain synchronization once the system isinitialized. At the transmitter-encoder end, a message is encoded usingthe chaotic dynamics simulated in the software, resulting in a digitalcontrol string that is sent to the receiver-decoder. Passing the controlstring through the same software (in reverse order) decodes the messageso that the receiver-decoder can output the plaintext message.

In a second software implementation of the present invention, mappingsinstead of differential equations can be used to define the chaoticsystems. In this case, the chaotic systems in the transmitter-encoderand receiver-decoder are defined to take an input value and produce anoutput value. The transmitter-encoder and receiver-decoder again holdduplicate copies of a file that contains all of the control informationthat is necessary to maintain synchronization once the system isinitialized. At the transmitter-encoder end, a message is encoded usingthe chaotic dynamics of the mapping function, resulting in a digitalcontrol string that is sent to the receiver-decoder. Passing the controlstring through the same software (in reverse order) decodes the messageso that the receiver-decoder can output the plaintext message.

The method and apparatus of the present invention can also beimplemented in hardware. The chaotic systems in the transmitter-encoderand receiver-decoder are still defined by a set of differentialequations, but these equations are then used to develop an electricalcircuit that will generate the same chaotic dynamics. The procedure forconversion of a differential equation into an equivalent circuit iswell-known and can be accomplished with operational amplifiers andmultipliers, as well as other devices known to one skilled in the art,configured with the proper feedbacks. The control information is storedin a memory device, and controls are applied by increasing voltage orinducing small current surges in the circuit. The control information isagain transmitted as a digital control string to the receiver-decoder,where the same controls are applied to an identical circuit so that themessage can be decoded and the plaintext message output.

In a second hardware implementation of the present invention, a mappingfunction is converted into a look-up table that can be stored on adigital memory chip, along with a table containing the controlinformation. A message is encoded by using the look-up table to generatethe chaotic dynamics, and the resulting control string is transmitted tothe receiver-decoder. Decoding is performed by reversing the process, sothat the controls are applied to the mapping function in the look-uptable to recover the chaotic dynamics and the original plaintextmessage.

The method and apparatus of the present invention can also beimplemented in lasers. In this implementation, a set of differentialequations is approximated using optical devices. Once the approximatesystem is developed, it defines the chaotic systems in thetransmitter-encoder and the receiver-decoder, and then control surfaces,partitions and microcontrols are defined for the chaotic dynamicsrealized by the laser system. The laser is driven into a chaotic mode ofoscillation, and controls are developed using, e.g. the occasionalproportional feedback (“OPF”) technique. [E.R. Hunt Phys. Rev. Lett. 67,1953 (1991)]. The control information is stored in a memory device thatcontains information defining the required controls for both the fullcontrols and the microcontrols, as described above. The microcontrolsare applied by using, e.g., OPF controls to drive the chaotic dynamicstoward the center of the partitions on the control surfaces. If themessage requires the system to be given a full control to shift into anearby partition, the magnitude of the OPF control would be larger. Inthis way, both the microcontrols and the full controls can be applied ina laser-based implementation. The control information is againtransmitted as a digital control stream to the receiver-decoder, wherethe same controls are applied to an identical laser-based system so thatthe message can be decoded and the plaintext message output.

It should be noted that the encoding in steps 22, 24 and 26 between asequence of message bits and the corresponding transmission bits ismany-to-one and one-to-many. In other words, a given sequence of messagebits can be encoded in many ways; similarly, a given sequence oftransmission bits can represent many different sequences of messagebits. It is only the dynamics of the first chaotic system in thetransmitter-encoder that allow the proper meaning to be discerned, andthe encoding is entirely dependent on the history of both the chaoticsystem and the message. FIG. 6 is a histogram of the mapping between4-bit messages and 4-bit transmissions. The histogram displays thefrequency of occurrence of each 4-bit transmission given itscorresponding 4-bit message. This shows that the present invention issomewhat analogous to a key-based encoding system in which the keychanges at each iteration, but the changes do not follow a prescribedpattern; rather, the key changes occur as a function of the message andthe history of the dynamics.

It should also be noted that calculations of the cross-correlationbetween the message bit stream and the transmission bit stream showsthere is essentially no correlation. Based on these and other similarresults, the transmission time series appears to be independent of themessage time series, and the transmission time series would look randomat the level of the autocorrelation function although other tests mightreveal non-random structure for certain embodiments.

Also, attempts have been made to find a way to take the transmission bitstream and construct some kind of dynamical model. In order to do this,sequences of bits from the transmission bit stream have been interpretedas integer digital numbers. For example, a sequence of bits such as 1100would be interpreted as the decimal number 12. By considering sequencesof bits, it is possible to consider statistical and dynamical tests fordeterminism. To do this, sequences of 4, 8, and 16 bits taken from thetransmitted signal have been considered. Consideration was given towhether the reconstructed data points taken from the disjoint timeseries will fill all possible positions in phase space. The results for2-dimensional reconstructions described below appear to hold in threedimensions as well. A calculation of reconstructed data points of theform x_(i)=(s_(i), s_(i+1)) where s_(i) is the decimal representation ofthe i-th disjoint 16-bit block shows that most grid points are coveredfor 16-bit sequences. Consequently, it does not appear possible to usereconstructions to find a distinguished subset of reconstructed pointsthat can be used to determine the state of the first chaotic system inthe transmitter-encoder.

Consideration was also given to whether there was a consistent patternto the dynamical evolution of the reconstructed points. Phase spacereconstructions were created to search for a consistent flow pattern,i.e. to look for some regularity to the plotted points or to thedynamical behavior as x₁→x₂→x₃ . . . If any predictable dynamicalbehavior were revealed, it might be possible to determine somethingabout the chaotic systems in the transmitter-encoder andreceiver-decoder. However, the flow patterns appeared as random linesconnecting the grid points so there is no dynamical information thatcould be gleaned from the transmitted data, and NLD forecasting wascompletely ineffective.

Thus, in summary, since the transmission bit stream is just a digitalsequence, there is no information that can be used to produce atime-delay phase space reconstruction in the usual sense. Consequently,the techniques that have been used to break chaotic communicationschemes are no longer applicable to this problem because there is noobvious way to extract geometric information from the transmittedsignal. Even from these preliminary tests, it appears that this binarychaotic communication system is much more difficult to analyze from anNLD perspective than earlier chaotic communication techniques thattransmitted a chaotic waveform.

An interesting perspective on the binary chaotic communication systemcan be gained by considering the first chaotic system in thetransmitter-encoder as a key generating device. In fact, as long as themicrocontrols are non-zero, the system will have only a finite number ofpossible trajectories, so it is fair to consider this as a keygeneration scheme. From this viewpoint, the interesting aspect of thisapproach is that the “key” would change at every iteration, but thechanges would not follow a predetermined functional pattern and would,in fact, be a function of the previous history of the chaos and themessage. This would alter the nature of a brute-force attack on thetransmission since it would make little sense to try all possible keyswhen the key changes at every iteration.

One important consideration in determining the security potential of acommunication scheme is that it is usually the case that the method ofcommunication is known, so the security must be in something likeprivate keys. For the digital chaotic communication system discussedhere, that would imply that the chaotic systems would be known (althoughit is arguable whether the intercepting party would need to know all ofthe operating parameters of the circuitry). If the chaotic systems inthe transmitter-encoder and receiver-decoder are known, the security inthis approach lies in the private function r(x), which could becalculated using any number of loops around the attractor, as well asthe perturbation rules stored in M. A brute-force approach to breakingthe transmission would be to calculate a set of functions r_(i)(x) wherei represents the number of loops that were used in the calculation.Then, the intercepting party could try each potential key functionsequentially until the message was decoded. Of course, the goal of thesystem designers must be to try to make this a difficult calculation,thereby achieving some degree of computational security.

Many applications of the present invention can be envisioned.in whicheither the authenticity or confidentiality (or both) of a message mustbe maintained: Examples of a few of the many applications are set forthbelow.

In an application in which it is important to broadcast communicationsthat are specific to one vehicle, such as air traffic control, thecurrent invention allows each vehicle to be given a different chaoticsystem. Once a message is encoded by a central broadcasting site usingthe chaotic system appropriate to a given user, the message can bedecoded only by that user. In a similar, but less-secure manner, asingle chaotic system can be used, with each vehicle being given adifferent initialization state. Only the vehicle with the correctinitialization state will be able to decode a message properly.

In an application in which there is a central broadcasting site servingmany users, such as a cellular phone network, each user can be given aunique chaotic system to encrypt all of its messages. When a user sendsout a message to another user, it passes through the central site, whereit is decoded and re-encoded before being transmitted. In this manner,all communications will be kept private.

In addition, if a user or group of users were given “smart cards” thatcontained a chip incorporating the present invention, it would bepossible for a central server to send initialization codes to a smartcard, resulting in the development of certain periodic orbits. If eachinitialization code is considered to be a query, and the resultingperiod of the orbit to be the answer, the central server can query auser until enough answers have been given to allow the user to bedistinguished from all other users (who would have different chaoticsystems and, hence, would give different answers). More importantly,since there is a large space of initialization codes to choose from, thecentral server could use different queries at each access, making itimpossible for a third party to intercept the responses from one sessionand use them for a new session.

The invention has been particularly shown and described above withreference to various preferred embodiments implementations andapplications. The invention is not limited, however, to the embodiments,implementations or applications described above, and modificationthereto may be made within the scope of the invention.

What is claimed is:
 1. A method for secure digital chaoticcommunication, said communication method comprising: a) a method ofencoding information, said encoding method comprising: obtaining amessage bit stream; applying a series of intermittent controls to afirst chaotic system to cause it to generate the message bit stream;creating a control/no control bit stream that is based on theapplication of the intermittent controls; and prepending asynchronization bit stream to the control/no control bit stream, tocreate a transmission bit stream from which the dynamics of the firstchaotic system cannot be reconstructed; b) transmitting the transmissionbit stream; c) a method of decoding information, said decoding methodcomprising: obtaining the transmission bit stream; extracting thesynchronization bit stream from the transmission bit stream and applyingthe synchronization bit stream to a second chaotic system to cause it tosynchronize with the first chaotic system; extracting the control/nocontrol bit stream from the transmission bit stream; and applyingintermittent controls, as indicated by the control/no control bitstream, to the second chaotic system causing it to generate the messagebit stream.
 2. The method for secure digital chaotic communication ofclaim 1, further comprising the steps of: a) repeating one or more timesthe method of encoding information therein; and b) repeating one or moretimes the method of decoding information therein.
 3. The method forsecure digital chaotic communication of claim 1 wherein the firstchaotic system is defined by a set of differential equations.
 4. Themethod for secure digital chaotic communication of claim 1 wherein thefirst chaotic system is defined by a mapping function.
 5. The method forsecure digital chaotic communication of claim 1 wherein the firstchaotic system is defined by an electrical circuit.
 6. The method forsecure digital chaotic communication of claim 1 wherein the firstchaotic system is defined by a configuration of optical devices.
 7. Asystem for secure digital chaotic communication, said communicationsystem comprising: d) a system for encoding information, said encodingsystem comprising: means for obtaining a message bit stream; means forapplying a series of intermittent controls to a first chaotic system tocause it to generate the message bit stream; means for creating acontrol/no control bit stream that is based on the application of theintermittent controls; and means for prepending a synchronization bitstream to the control/no control bit stream, to create a transmissionbit stream; e) system for transmitting the transmission bit stream,wherein no chaotic signals are transmitted; f) system for decodinginformation, said decoding system comprising: means for obtaining thetransmission bit stream; means for extracting the synchronization bitstream from the transmission bit stream and applying the synchronizationbit stream to a second chaotic system to cause it to synchronize withthe first chaotic system; means for extracting the control/no controlbit stream from the transmission bit stream; and means for applyingintermittent controls, as indicated by the control/no control bitstream, to the second chaotic system causing it to generate the messagebit stream.
 8. The system for secure digital chaotic communication ofclaim 7 wherein the first chaotic system is defined by a set ofdifferential equations.
 9. The system for secure digital chaoticcommunication of claim 7 wherein the first chaotic system is defined bya mapping function.
 10. The system for secure digital chaoticcommunication of claim 7 wherein the first chaotic system is defined byan electrical circuit.
 11. The system for secure digital chaoticcommunication of claim 7 wherein the first chaotic system is defined bya configuration of optical devices.
 12. A system for secure digitalchaotic communication, said communication system comprising: a source ofa message bit stream; a transmitter-encoder to apply a series ofintermittent controls to a first chaotic system to cause it to duplicatethe message bit stream, to generate a control/no control bit stream thatis based on the application of the intermittent controls and to prependa synchronization bit stream to the control/no control bit stream; thetransmitter-encoder to transmit the control/no control bit stream andprepended synchronization bit stream, wherein the message bit stream isnot transmitted; a receiver-decoder to receive the control/no controlbit stream and prepended synchronization bit stream; and a secondchaotic system, to which first the synchronization bit stream is appliedand then the series of intermittent controls is applied to cause it toduplicate the message bit stream.